I'm looking for a solution that doesn't use a loud relay (click) to make a simple dry contact closure. I've soldered wires to the circuit board of an existing product. I was curious if I could bypass the use of a relay altogether and use the digital-outs of the adruino and pulsed the outputs if the current would be enough to simulate a contact closure.
Yes and no. The Arduino's outputs can only source or sink around 20~30mA at 5V, which is not enough to actuate most relay coils. Could be you need a transistor. But can't say anything more certain because of the total lack of relevent detail of the components in your question.